How to apply a specific modifier on selected objects?
Blender 2.79

This should work:

import bpy

for ob in bpy.context.selected_objects:
    bpy.context.scene.objects.active = ob
    for mod in [m for m in ob.modifiers if m.type == 'ARMATURE']:
        bpy.ops.object.modifier_apply( modifier = mod.name )


You may have some warnings though (like via the UI).

Blender 2.8

Use view_layer instead of scene:

import bpy

for ob in bpy.context.selected_objects:
    bpy.context.view_layer.objects.active = ob
    for mod in [m for m in ob.modifiers if m.type == 'ARMATURE']:
        bpy.ops.object.modifier_apply( modifier = mod.name )